What makes a polygon regular
A polygon is regular when every side is the same length and every interior angle is equal. A regular three-sided polygon is an equilateral triangle, a regular four-sided one is a square, and as the side count rises the shape looks more and more like a circle.
That symmetry means a single side length, together with the number of sides, pins down every other measurement of the figure.
What each output means
The calculator reports the full set of standard measurements:
- Perimeter: the side length multiplied by the number of sides.
- Interior angle: the angle at each corner, (n−2)·180°/n.
- Exterior angle: the turn between sides, 360°/n.
- Apothem: the perpendicular distance from the centre to the middle of a side.
- Circumradius: the distance from the centre out to a vertex.
- Area: the space enclosed, ¼·n·s²·cot(π/n).
Why the angles work the way they do
The interior angles of any polygon sum to (n−2)·180°, so each angle of a regular polygon is that total divided evenly across the n corners. The exterior angles always sum to a full 360° no matter how many sides, which is why each exterior angle is simply 360°/n.
Caveats
A polygon needs at least three sides, so n must be a whole number of 3 or more, and the side length must be positive. This tool covers regular (equal-sided) polygons only; irregular shapes need their vertices specified individually.
Formula
area = ¼·n·s²·cot(π/n); interior angle = (n−2)·180°/nFrequently asked questions
- What is the apothem used for?
- The apothem is the centre-to-edge distance. Area also equals ½ × perimeter × apothem, which is a handy alternative to the cotangent formula.